• In statistics, point estimation refers to the process of using sample data to calculate the point estimate (single value).
• It’s a “best guess” for an unknown population parameter
• The type of point estimate (sample mean, sample proportion, etc.) varies based on the population parameter examined
• Requires a point estimator: mathematical formula that turns raw data into a point estimate
• Point estimators must be valid, leading one to consider: What are properties of a good point estimator?
  • Unbiased: expected value should be equal to the true value of the population estimator
  • Efficiency: point estimator should have the smallest variance
  • Consistentcy: gets closer to the true value of the population parameter as the sample size increases

• Parameter: Characteristic of the entire population and the unknown value we wish to estimate
• Statistic: Characteristic of sample mean
• Point Estimate: The single value calculated from the sample data using the estimator
• Point Estimator (re-iterated): formula or rule that is used on sample data to guess the parameter
• Random sampling: point estimation relies on a randomly collected sample data, otherwise inferences about the population cannot be made

library(plotly)
set.seed(42)
# Initialize values for random simulation of height frequencies with a known true population mean
population_mean <- 172
population_sd <- 10
population_data <- rnorm(50000, mean = population_mean, sd = population_sd)

sample_size <- 100
sample_data <- sample(population_data, size = sample_size)

sample_mean_estimate <- mean(sample_data)

p <- plot_ly(x = population_data, type = "histogram", nbinsx = 50, name = "Population") %>%
# Create lines for the true population mean and point estimate in the plot
  add_lines(
    x = ~population_mean, y = 0, name = paste("True Population Mean (μ) =", round(population_mean, 2)),
    line = list(color = 'orange', width = 4, dash = 'dash')
  ) %>%
  add_lines(
    x = ~sample_mean_estimate, y = 0, name = paste("Point Estimate (x̄) =", round(sample_mean_estimate, 2)),
    line = list(color = 'red', width = 4)
  )

p %>% 
  layout(
    title = list(text = "<b>Frequency of Height</b>", x = 0.5, xanchor = "center"),
    xaxis = list(title = "<b>Height (cm)</b>"),
    yaxis = list(title = "<b>Frequency</b>"),
    shapes = list(
      list(
        type = 'line', x0 = population_mean, x1 = population_mean,
        y0 = 0, y1 = 1, yref = 'paper', 
        line = list(color = 'orange', width = 4, dash = 'dash')
      ),
      list(
        type = 'line', x0 = sample_mean_estimate, x1 = sample_mean_estimate,
        y0 = 0, y1 = 1, yref = 'paper',
        line = list(color = 'red', width = 4)
      )
    )
  )

• For the plotly plot, the population parameter was the true population mean, so it made the most sense to calculate the sample mean as the point estimate.
• This was our point estimator:

\[ \color{black}{\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}} \] -\(\bar{x}\) is the sample mean (our point estimate)

-\(\sum x_i\) is the sum of all the individual values in our sample

-\(n\) represents the sample size

• Sample Variance Formula (ggplot 1) \[ \color{black}{s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} \]
• Sample Variance Formula (ggplot 2) \[ \color{black}{s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}} \] \[\begin{align} \text{Where:} \\ s^2 &= \text{sample variance} \\ s &= \text{sample standard deviation} \\ x_i &= \text{individual data point (i-th observation)} \\ \end{align} \]